Two Classes of Infrasoft Separation Axioms

One of the considerable topics in the soft setting is the study of soft topology which has enticed the attention of many researchers. To contribute to this scope, we devote this work to investigate two classes of separation axioms with respect to the distinct ordinary elements through one of the generalizations of soft topology called infrasoft topology. We first formulate the concepts of infratp-soft Tj using total belong and partial nonbelong relations and then introduce the concepts of infra-tt-soft Tj-spaces using total belong and partial nonbelong relations. To illustrate the relationships between them, we provide some examples. We discuss their fundamental properties and study their behaviors under some special types of infrasoft topologies. An extensive discussion is given for the transmission of these two classes between infrasoft topology and its parametric infratopologies. In the end, we demonstrate which ones have topological and hereditary properties, and we show their behaviors under the finite product of soft spaces.


Introduction
One of the powerful mathematical approaches to cope with uncertain problems is the soft set. It was proposed in 1999 by Molodtsov [1] who explained its applications to different areas such as smoothness of functions, theory of measurement, game theory, Riemann integration, operations research, etc. In 2002, soft sets were successfully applied to handle decision-making problems by Maji et al. [2]. ey introduced some operations and operators on soft sets such as their soft intersection and union and the complement of a soft set in [3]. Despite the weakness of some concepts and results in these early references, they form the essential start point of soft set theory. To keep some properties and results of crisp set theory, Ali et al. [4] displayed new types of these operations and operators. Novel types of them were established such as lower and upper soft equality [5], gf-soft union [6], and T-soft subset and T-soft equality [7]. Since the advent of soft sets, many authors applied successfully to address problems in some disciplines such as computer science [8], decision-making [9,10], and medical science [11,12]. ese applications prove the adequacy of soft sets to treat and model a lot of real-life issues.
In 2011, Shabir and Naz [13] defined soft topologies as hybridization of classical topology and soft sets. en, numerous researchers have discussed the topological concepts, properties, and results in soft topologies; see [14][15][16][17]. Soft point [18,19] is an essential concept in soft set theory; it represents the soft version of an ordinary element. Some types of soft topologies called enriched and extended soft topologies are studied [19].
Some topological properties are kept under conditions weaker than topology's conditions; also, some phenomena are described under structures relaxing a topology. e structures of suprasoft topology [20] and infrasoft topology [21] have been born with these goals. ey have become two of the most interesting developments of soft topology in recent years. Supratopology is a class of subsets that extend the concept of topological space by dispensing with the postulate that the class is closed under finite intersections, whereas infratopology is a class of subsets that extend the concept of topological space by dispensing with the postulate that the class is closed under arbitrary unions. Generalized soft topology [22] and soft weak structure are other generalizations of soft topology. Recently, the concepts of compactness and connectedness have been studied through the frame of infrasoft topology in [23,24], respectively.
Another technique of generalizing soft topology was introduced by combining other structures with a soft topology. In this regard, Ittanagi [25] initiated the concept of soft bitopology.
Our contribution to this field concerns the analysis of what type of "separation axioms" are meaningful in the study of infrasoft topology. As was the case of classical topology, soft separation axioms are among the most interesting and substantial ideas in soft topology. ey form a tool to establish more restricted (and wider) classes of well-behaved soft spaces. It should be noted that a large variety of separation axioms in the soft structures is attributed to two factors. One is the distinguished objects that we intend to separate: they can be either ordinary points or soft points. e other is the type of belongingness and nonbelongingness relations that we require in the definitions: they can be either partial or total. For more details about this subject, see [26], which investigated new sorts of separation axioms on suprasoft topologies, and [10,27], which studied two kinds of soft T i -spaces on soft topologies. In [28], the researchers presented soft separation axioms with respect to distinct soft points. Terepeta [29] discussed soft separation axioms using their counterparts in classical topological spaces. Singh and Noorie [30] conducted a comparative study among some soft separation axioms published in the literature.
In this article, we note that many properties of soft topological spaces are still valid on infrasoft topological spaces and initiating examples that show some relationships between certain topological concepts is easier on infrasoft topological spaces. erefore, we aim in this paper to perform an exhaustive analysis of infrasoft topological spaces. e organization of this manuscript is as follows. Section 2 mentions some definitions and properties related to soft set theory and infrasoft topology. Section 3 initiates the concepts of infra-tp-soft T i -spaces (i � 0, 1, 2, 3, 4) and studies basic properties. Section 4 defines the concepts of infratt-soft T i -spaces (i � 0, 1, 2, 3, 4) and discusses main properties. Elucidative examples are supplied to validate the obtained relationships and results. Ultimately, Section 5 outlines the paper's achievements and proposes some future works.

Preliminaries
e concepts and results that we need in this paper are recalled in this section.

Soft Set eory.
e concepts and results of soft set theory that we need in this paper are mentioned in this part.
Definition 1 (see [1]). A map E from Σ which is set of parameters to the power set 2 Y of Y is called a soft set over Y . It is denoted by E Σ and identified as E Σ � (σ, E(σ)): σ ∈ Σ and E(σ) ∈ 2 Y . e set of all soft sets over Y with Σ as a set of parameters is symbolized by S(Y Σ ).
Definition 2 (see [4]). e complement of a soft set E Σ , symbolized by E c Σ , is given by Definition 3 (see [2]). If the image of each parameter of Σ under a map E: Σ ⟶ Y is the empty set, then E Σ is called the null soft set over Y . Its complement is called the absolute soft set. e notations Φ and Y , respectively, denote the null and absolute soft sets.
Definition 4 (see [18,19]). If the image of one parameter, say σ, under a map P: Σ ⟶ Y is a singleton set, say ] { }, and the image of each parameter σ ′ ∈ Σ, σ { } is the empty set, then a soft set P Σ is called a soft point over Y . It is briefly symbolized by P ] σ .
Definition 5 (see [4]). e intersection of two soft sets E Ω and F Γ over Y , symbolized by Definition 6 (see [4]). e union of two soft sets E Ω and F Γ over Y , symbolized by E Ω ∪ F Γ , is a soft set H Σ , where Σ � Ω ∪ Γ and a map H: Σ ⟶ 2 Y is given as follows: Definition 7 (see [31] Definition 8 (see [13]). A soft set E Σ over Y which is defined by E(σ) � ] { } for each σ ∈ Σ is called a singleton soft set. It is denoted by ] Σ .
Definition 9 (see [27]). If all components of a soft set are equal, then we called it a stable soft set.
e Cartesian product of E Σ and H Ω , symbolized by (G × H) Σ×Ω , is defined as e definition of soft maps given in [32] was reformulated in a way that reduces calculation burden and gives a justification (logical explanation) for some soft concepts such as why we determine that f η is injective or surjective according to its two crisp maps f and η.
Definition 11 (see [33]). Let f: X ⟶ Y and η: Σ ⟶ Ω be two crisp maps. A soft map f η of S(X Σ ) into S(Y Ω ) is a relation such that each soft point in S(X Σ ) is related to one and only one soft point in Definition 12 (see [32]). A soft map f η : S(Y Σ ) ⟶ S(Z Ω ) is said to be injective (resp. surjective, bijective) if both f and η are injective (resp., surjective, bijective).
Definition 13 (see [13,27]). For a soft set E Σ over Y and ] ∈ Y , we say that Proposition 1 (see [27]). Consider a soft map f η : S(Y Σ ) ⟶ S(Z Ω ) and let E Σ and H Ω be soft sets in S(Y Σ ) and S(Z Ω ), respectively. en the next results hold true:

Infrasoft Topological Spaces.
e concepts and results of infrasoft topology that we need in this paper are mentioned in this part.
Definition 14 (see [21]). e collection δ of soft sets over Y under a parameters set Σ is said to be an infrasoft topology on Y if it is closed under finite soft intersection and Φ ∈ δ. e triple (Y , δ, Σ) is called an infrasoft topological space. Every member of δ is called an infrasoft open set and its relative complement is called an infrasoft closed set. We called (Y , δ, Σ) stable if all its infrasoft open sets are stable.
Proposition 3 (see [21]). Suppose that δ σ σ∈Σ is a family of crisp infratopologies on Y . en the family defines an infrasoft topology on Y .
e infrasoft topological space given in the above proposition is called an extended infrasoft topology.

Infra-tp-Soft T i -Spaces
In this section, we display new family of separation axioms, namely, infra-tp-soft T i -spaces (i � 0, 1, 2, 3, 4), where t and p are the first letters of total belong and partial nonbelong relations that are used to define these spaces. We explore main properties and supply various examples to elucidate the relationships between them.
Before we investigate the relationships between the above infrasoft axioms, we need the following auxiliary result.

Lemma 1. Every infrasoft open (infrasoft closed) subset of an infra-tp-soft regular space is
this implies we cannot find an infrasoft open set whose intersection with F Σ is the null soft set such that ] totally belongs to it. Hence, E Σ must be stable.
One can similarly prove the lemma for an infrasoft closed set.
e proof is clear for the cases i � 1, 2. To prove the case of i � 3, let ], μ be two distinct elements in an infra- It follows from the above lemma that μ / ⋐ E Σ . en μ ∈ U Σ ; hence, we obtain the desired result.
Examples below explain that we cannot reverse the above proposition in general.
Obviously, (Y , δ 1 , Σ), (Y , δ 2 , Σ), and (Y , δ 3 , Σ) are, respectively, infra-tp-soft T 0 , infra-tp-soft T 1 , and infratp-soft T 2 . On the other hand, (Y , δ 1 , Σ) is not infra-tp-soft Remark 1. In the above example, (Y , δ 3 , Σ) is an infratp-soft T 4 -space. As we know, the soft topology and general topology (infrasoft topology and general infratopology) are identical if a set of parameters is a singleton. en we can say that there is an infrasoft topology which is infra-tp-soft T 3 , but not infra-tp-soft T 4 . Hence, the concepts of infra-tp-soft T 3 and infra-tp-soft T 4 -spaces are independent of each other. Now, we explore and discuss some properties of infratp-soft T i -spaces.
is ends the proof that is an infra-tp-soft regular space, then the following concepts are identical.
Proof. We prove the proposition using mathematical induction. When |Y | � 2, we need two infrasoft open sets beside the null and absolute soft sets. en the claim is true for i � 2. Suppose that the claim is true for |Y | � n; i.e., |δ| � n + 2. Without loss of generality, consider (Y , δ, Σ) as the smallest infra-tp-soft T 2 -space. Now, let |Y | � n + 1; i.e., { } is the smallest infra-tp-soft T 2 -space on Y ′ . Hence, we obtain the desired result.
In the following result we will study the properties and relationships between infrasoft topology and its parametric infratopologies. δ, Σ). en there exist disjoint infrasoft open sets E Σ and F Σ totally containing ] and μ, respectively. erefore, E(σ) and F(σ) are disjoint infraopen sets containing ] and μ, respectively. Hence, (Y , δ σ ) is an infra-T 2 -space.
To prove the proposition in case of i � 3, it suffices to prove the axiom of infraregularity. To do that, let G be an infraclosed subset of (Y , δ σ ) such that ] ∉ G. Since δ is infrasoft tp-regular, a soft set H Σ � (σ, G): for each σ ∈ Σ { } is an infrasoft closed subset of (Y , δ, Σ) such that ] ∉ H Σ . By assumption, there exist disjoint infrasoft open sets E Σ and F Σ such that H Σ ⊆ E Σ and ] ∈ F Σ .
To demonstrate that the converse of the proposition above fails, we supply the next example.
□ Example 2. Consider Y and Σ as the same as in Example 1.
From the above example and the next two examples, we elucidate that there is not a relationship between infratp-soft T i -spaces and their parametric infratopological spaces when i � 0, 1, 4.

Journal of Mathematics
Proof. We only prove the theorem when i � 4. e case of i � 1 was proved in the above theorem. So, it suffices to prove the axioms of infrasoft normality. To do this, let H Σ , L Σ be two disjoint infrasoft closed sets. en H(σ) and L(σ) are two disjoint infraclosed sets for each σ ∈ Σ. Since (X, δ σ ) is infranormal, there are two disjoint infraopen sets U σ and V σ such that H(σ)⊆U σ and L(σ)⊆V σ .
Proof. We give a proof for i � 3. e other cases are made similarly.
First, we prove that Second, we prove that (Z, δ Z , Σ) is infra-tp-soft regular. To do this, let L Σ be an infrasoft closed subset of (Z, δ Z , Σ) and μ ∈ Z such that μ ∉ L Σ . en there exists an infrasoft closed subset H Σ of (Y , δ, Σ) such that L Σ � Y ∩ H Σ . By assumption, there exist disjoint infrasoft open sets E Σ and F Σ such that H Σ ⊆ E Σ and μ ∈ F Σ . Now, us, (Z, δ Z , Σ) is infra-tp-soft regular. is ends the proof that (Z, δ Z , Σ) is infra-tp-soft T 3 . Following similar technique given above, one can prove the following result.

Theorem 8. Every infrasoft closed subspace
e concepts of infra-tp-soft T i -spaces are preserved under finite product spaces, where i � 0, 1, 2.
Proof. We give a proof for the theorem when j � 2. e remaining two cases follow similar lines.
Without loss of generality, we consider (Y 1 , δ 1 , Σ 1 ) and By assumption, δ 1 contains two disjoint infrasoft open sets E Σ 1 , F Σ 1 such that ] 1 ∈ E Σ 1 and ] 2 ∉ E Σ 1 , and and We close this section by studying the behaviors of infratp-soft T i -spaces under some types of soft maps.

Proposition 10. e inverse image of infra-tp-soft T i -spaces under an infrasoft continuous map f η is infra-tp-soft T i for
i � 0, 1, 2 provided that f is injective and η is surjective.
By the injectivity of f, there are only two distinct points ζ, ξ ∈ Z such that f(]) � ζ and f(μ) � ξ. By assumption, there are two disjoint infrasoft open sets are disjoint infrasoft open sets. By the surjectivity of η we obtain from . Hence, we get the coveted result.
In a similar manner, one can prove the next results. □ 0, 1, 2, 3, 4) is preserved under an infrasoft homeomorphism map. 1,2,3,4) In this part, we apply the relations of total belong and total nonbelong to define the concepts of infra-tt-soft T i -spaces (i � 0, 1, 2, 3, 4), where tt is the abbreviation for the first letter of total belong and total nonbelong relations that are used to define these spaces. We provide some counterexamples to illustrate the relationships between these concepts and the interrelations between them and infra-tp-soft T i -spaces. (v) An infra-tt-soft T 3 (resp., infra-tt-soft T 4 ) -space if it is infra-tt-soft regular (resp., infrasoft normal) and infra-tt-soft T 1 .

Infra-tt-Soft T i -Spaces
is section begins with the next result which describes the relationships between infra-tt-soft T i -spaces as well as the interrelations between them and infra-tp-soft T i -spaces.

Proposition 14
( (iv) It is clear that every infra-tp-soft regular space is infra-tt-soft regular. According to Lemma 1, the concepts of infra-tt-soft T 1 -space and infra-tp-soft T 1 -space are identical. Hence, we get the coveted result.
To elucidate that the converses of the above proposition fail, we provide the next three counterexamples. □ Example 5. Let Y be any infinite set and ], μ ∈ Y . We define the following three infrasoft topologies on Y with Σ as set of parameters such that |Σ| ≠ 1.
Example 7. Any discrete soft topological space is infratt-soft T 3 . On the other hand, it contains some infrasoft open sets which are not stable, so it is not infra-tp-soft regular. Hence, it is not infra-tp-soft T 3 .
Remark 3. We show that (Y , δ 3 , Σ) given in Example 5 is not infra-tt-soft T 3 . In contrast, one can check that it is infratt-soft T 4 . According to Remark 1 there is an infra-tt-soft T 3 -space which is not infra-tt-soft T 4 . Hence, the concepts of infra-tt-soft T 3 and infra-tt-soft T 4 -spaces are independent of each other.
Proof. It comes from the fact that partial nonbelong and total nonbelong relations are identical with respect to the stable soft sets.
In what follows, we study and investigate some properties of infra-tt-soft T i -spaces. □ Proposition 15. If every singleton soft subset of (Y , δ, Σ) is infrasoft closed, then (Y , δ, Σ) is an infra-tt-soft T 1 -space.
is infra-tt-soft T 1 . e converse of the above proposition is not always true as the next counterexample explains. □ Example 8. Let δ � N ∪ F Σ ⊏N: F Σ is finite be an infrasoft topology on the set of natural numbers N with Σ as any set of parameters. It is clear that (N, δ, Σ) is infra-tt-soft T 1 . But every singleton soft set is not infrasoft closed.

Proposition 21.
e inverse image of infra-tt-soft T i -spaces under an infrasoft continuous map f η is infra-tt-soft T i for i � 0, 1, 2 provided that f is injective.
In the end of this work, we summarize the relationships between the two introduced infrasoft separation axioms in Figure 1.

Conclusion and Future Work
In this paper, we have introduced two types of separation axioms in the frame of infrasoft topological spaces with respect to distinct ordinary points. e reasons beyond this study are the following.
(1) Construct new classes of soft spaces and explore the (classical) topological properties that are valid in these spaces. (2) Investigate these soft spaces in the frame of rough sets models as those given [34][35][36] and show whether they have an influence on the rough approximations and accuracy measures. (3) Obtain new results induced from the interaction between these soft spaces and some celebrated concepts such as infrasoft connectedness and infrasoft compactness which we will introduce later.
We outline the achievements of this work in the following.
(1) Formulate the concepts of infra-tp-soft T i -spaces for i � 0, 1, 2, 3, 4 using total belong and partial nonbelong relations. (2) Formulate the concepts of infra-tt-soft T i -spaces for i � 0, 1, 2, 3, 4 using total belong and total nonbelong relations. (3) Provide some counterexamples to show the relationships between the spaces in each type of them as well as the interrelationships between the two types. (4) Study the transmission property of these two types of soft spaces between infrasoft topology and its parametric infratopologies. (5) Discuss the image and preimage of these two types of soft spaces under some infrasoft maps.
In upcoming papers, we plan to do the following.

Data Availability
No data were used to support this study.  Journal of Mathematics 9